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If a polynomial function [tex]\( f(x) \)[/tex] has roots [tex]\(-9\)[/tex] and [tex]\(7-i\)[/tex], what must be a factor of [tex]\( f(x) \)[/tex]?

A. [tex]\((x - (7+i))\)[/tex]
B. [tex]\((x - (-7-i))\)[/tex]
C. [tex]\((x + (7+i))\)[/tex]
D. [tex]\((x + (7-i))\)[/tex]

Sagot :

GinnyAnswer
To determine a factor of a polynomial given its roots, let's analyze the following:

1. Given Roots:
- One root is [tex]\(-9\)[/tex].
- Another root is [tex]\(7 - i\)[/tex].

2. Complex Conjugate Root Theorem:
- If the coefficients of the polynomial are real, then complex roots occur in conjugate pairs. Therefore, if [tex]\(7 - i\)[/tex] is a root, then [tex]\(7 + i\)[/tex] must also be a root.

3. Constructing Factors:
- For a root [tex]\(a\)[/tex], the corresponding factor of the polynomial is [tex]\((x - a)\)[/tex].

4. Factors Based on Given Roots:
- The root [tex]\(-9\)[/tex] corresponds to the factor [tex]\((x - (-9))\)[/tex] or [tex]\((x + 9)\)[/tex].
- The root [tex]\(7 - i\)[/tex] corresponds to the factor [tex]\((x - (7 - i))\)[/tex].
- The root [tex]\(7 + i\)[/tex] corresponds to the factor [tex]\((x - (7 + i))\)[/tex].

Given the possible answers, we need to identify which explicitly matches the form of one of our derived factors for these roots:

- [tex]\((x - (7 + i))\)[/tex]: This matches the factor corresponding to the root [tex]\(7 + i\)[/tex].
- [tex]\((x - (-7 - i))\)[/tex]: This does not match any factor related to our given roots.
- [tex]\((x + (7 + i))\)[/tex]: This is not correctly formed, as it does not match the factor form [tex]\((x - (7 + i))\)[/tex] corresponding to the root [tex]\(7 + i\)[/tex].
- [tex]\((x + (7 - i))\)[/tex]: This is not correctly formed, as it does not match the factor form [tex]\((x - (7 - i))\)[/tex] corresponding to the root [tex]\(7 - i\)[/tex].

Therefore, the correct factor of [tex]\(f(x)\)[/tex] must be [tex]\((x - (7 + i))\)[/tex].

Thus, the correct factor is [tex]\((x - (7 + i))\)[/tex].
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